Problem 21 - Algebra 2

Question

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rr}{-4} & {6} \\ {6} & {-9}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrix does not have an inverse because its determinant is zero.

Step 1: Understand Inverse Matrix

A matrix \( A \) has an inverse if there exists a matrix \( A^{-1} \) such that \( A \cdot A^{-1} = I \), where \( I \) is the identity matrix. For a 2x2 matrix \( A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \), the inverse is given by \( A^{-1} = \frac{1}{ad-bc} \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right] \), provided \( ad-bc eq 0 \).

Step 2: Calculate the Determinant

Calculate the determinant of the matrix \( A = \left[\begin{array}{rr}{-4} & {6} \ {6} & {-9}\end{array}\right] \). The determinant is found using the formula \( ad-bc \). For our matrix, \( a = -4 \), \( b = 6 \), \( c = 6 \), and \( d = -9 \). Thus, the determinant \( ad-bc = (-4)(-9) - (6)(6) = 36 - 36 = 0 \).

Step 3: Determine if the Inverse Exists

Since the determinant calculated in Step 2 is 0, an inverse does not exist for the matrix. A matrix has an inverse only if its determinant is non-zero.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix, this number plays a crucial role in understanding the matrix's properties, particularly its invertibility. For the matrix \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \], the determinant \( det(A) \) is found using the formula:

• \( ad-bc \)

The determinant helps tell us if a matrix has an inverse. If the result is zero, the matrix cannot be inverted. In the example given where the matrix is \[ \begin{bmatrix} -4 & 6 \ 6 & -9 \end{bmatrix} \], the determinant calculation came out to be 0. This means the matrix doesn't have an inverse.

Identity Matrix

The identity matrix is a key concept in matrix algebra. It acts like the number "1" for matrices. When a matrix is multiplied by the identity matrix, it remains unchanged. The identity matrix for a 2x2 matrix is represented as:

• \[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]

If a matrix \( A \) has an inverse, multiplying it by its inverse gives the identity matrix, i.e., \( A \cdot A^{-1} = I \). This property is fundamental to the concept of matrix inverses and is essential when solving systems of linear equations or for transforming spaces in more advanced mathematics.

2x2 Matrix

A 2x2 matrix is a simple matrix structure that consists of two rows and two columns. It can be written as:

• \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]

Each element \( a, b, c, \) and \( d \) are numbers that define the matrix's properties and behavior. 2x2 matrices are often used in basic linear algebra because they are straightforward to work with due to their small size, yet they still include essential matrix properties such as determinants and eigenvalues. They are a perfect starting point for learning techniques that apply to larger matrices.

Matrix Algebra

Matrix algebra is an essential branch of mathematics that deals with operations involving matrices. This includes addition, subtraction, multiplication, and finding inverses. These operations are vital for solving linear equations and for applications in various fields such as physics, computer science, and engineering. To multiply matrices, the number of columns in the first matrix must match the number of rows in the second. The multiplication is done by taking the dot product of rows from the first and columns from the second matrix. Understanding matrix algebra allows for efficient computations and sophisticated transformations in space, making it an invaluable tool in both theoretical and applied mathematics.